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Dynamics in One Non-Archimedean Variable

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The theory of complex dynamics in one variable, initiated by Fatou and Julia in the early twentieth century, concerns the iteration of a rational function acting on the Riemann sphere. Building on foundational investigations of $p$-adic dynamics in the late twentieth century, dynamics in one non-archimedean variable is the analogous theory over non-archimedean fields rather than over the complex numbers. It is also an essential component of the number-theoretic study of arithmetic dynamics. This textbook presents the fundamentals of non-archimedean dynamics, including a unified exposition of Rivera-Letelier's classification theorem, as well as results on wandering domains, repelling periodic points, and equilibrium measures. The Berkovich projective line, which is the appropriate setting for the associated Fatou and Julia sets, is developed from the ground up, as are relevant results in non-archimedean analysis. The presentation is accessible to graduate students with only first-year courses in algebra and analysis under their belts, although some previous exposure to non-archimedean fields, such as the $p$-adic numbers, is recommended. The book should also be a useful reference for more advanced students and researchers in arithmetic and non-archimedean dynamics.
Robert L. Benedetto, Amherst College, MA.
Introduction Basic dynamics on $\mathbb{P}^1(K)$ Some background on non-archimedean fields Power series and Laurent series Fundamentals of non-archimedean dynamics Fatou and Julia sets The Berkovich projective line Rational functions and Berkovich space Introduction to dynamics on Berkovich space Classifying Berkovich Fatou components Further results on periodic components Wandering domains Repelling points in Berkovich space The equilibrium measure Proofs of results from non-archimedean analysis Proofs of Berkovich space results Proofs of results on Berkovich maps Fatou components without Berkovich space Other constructions of Berkovich spaces Bibliography Index.
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